We discuss the building blocks of a universal quantum computer within the circuit model of computation and how this is implemented using superconducting quantum circuits. In particular, we discuss, one by one, the creation of quantum registers, resetting of quantum bits, qubit measurements, single-qubit operations, and universal two-qubit gates, and how these are all implemented using the tools from earlier chapters. We discuss how to calibrate the errors in the qubits and in the operations, assigning them complete descriptions via positive maps. We explain how these errors can be corrected and how to implement a fault-tolerant quantum computer, focusing on the paradigm of stabilizer codes and the surface code in particular. We close with a discussion on the outlook for quantum computers in the near term and the NISQ paradigm of computation.

1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation.

2. Introduce the Gross-Pitaevskii equation and its applicable conditions.

3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap.

4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior.

5. Introduce the criterion for superfluidity and determine the superfluid critical velocity.

6. Introduce two-fluid hydrodynamics.

7. Introduce the Bogoliubov theory.

8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation.

9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate.

10. Summarize the difference between an interacting and non-interacting Bose condensate.

11. Introduce the Bragg spectroscopy.

12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence.

13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit.

14. Introduce scale invariant quantum gases.

15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence.

16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state.

17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state.

18. Discuss interference effects in density and density-density correlations.

19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.

This chapter is intended to introduce some basic notions of quantum theory needed in the subsequent chapters for the reader who is not familiar with them. Quantum mechanics is a fundamental physics subject that studies the phenomena at the atomic and subatomic scales. This chapter introduces the required mathematical tools and presents the postulates mainly through their mathematical formalisms. The physics interpretation of these is only very briefly discussed.